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Properties

Label 2-3e6-243.103-c1-0-23
Degree 22
Conductor 729729
Sign 0.926−0.375i0.926 - 0.375i
Analytic cond. 5.821095.82109
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.02 + 0.562i)2-s + (2.05 + 1.24i)4-s + (3.69 − 0.143i)5-s + (0.392 + 0.0613i)7-s + (0.582 + 0.617i)8-s + (7.55 + 1.78i)10-s + (−0.272 − 1.99i)11-s + (−0.138 + 0.202i)13-s + (0.758 + 0.344i)14-s + (−1.41 − 2.67i)16-s + (−3.38 + 1.69i)17-s + (0.121 + 2.08i)19-s + (7.78 + 4.29i)20-s + (0.572 − 4.18i)22-s + (−0.303 + 0.785i)23-s + ⋯
L(s)  = 1  + (1.42 + 0.397i)2-s + (1.02 + 0.621i)4-s + (1.65 − 0.0641i)5-s + (0.148 + 0.0231i)7-s + (0.206 + 0.218i)8-s + (2.38 + 0.565i)10-s + (−0.0821 − 0.601i)11-s + (−0.0385 + 0.0561i)13-s + (0.202 + 0.0921i)14-s + (−0.352 − 0.669i)16-s + (−0.820 + 0.411i)17-s + (0.0278 + 0.479i)19-s + (1.74 + 0.960i)20-s + (0.121 − 0.892i)22-s + (−0.0632 + 0.163i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.926−0.375i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.926−0.375i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.926−0.375i0.926 - 0.375i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(685,⋅)\chi_{729} (685, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.926−0.375i)(2,\ 729,\ (\ :1/2),\ 0.926 - 0.375i)

Particular Values

L(1)L(1) ≈\approx 3.88301+0.756902i3.88301 + 0.756902i
L(12)L(\frac12) ≈\approx 3.88301+0.756902i3.88301 + 0.756902i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
good2 1+(−2.02−0.562i)T+(1.71+1.03i)T2 1 + (-2.02 - 0.562i)T + (1.71 + 1.03i)T^{2}
5 1+(−3.69+0.143i)T+(4.98−0.387i)T2 1 + (-3.69 + 0.143i)T + (4.98 - 0.387i)T^{2}
7 1+(−0.392−0.0613i)T+(6.66+2.13i)T2 1 + (-0.392 - 0.0613i)T + (6.66 + 2.13i)T^{2}
11 1+(0.272+1.99i)T+(−10.5+2.94i)T2 1 + (0.272 + 1.99i)T + (-10.5 + 2.94i)T^{2}
13 1+(0.138−0.202i)T+(−4.68−12.1i)T2 1 + (0.138 - 0.202i)T + (-4.68 - 12.1i)T^{2}
17 1+(3.38−1.69i)T+(10.1−13.6i)T2 1 + (3.38 - 1.69i)T + (10.1 - 13.6i)T^{2}
19 1+(−0.121−2.08i)T+(−18.8+2.20i)T2 1 + (-0.121 - 2.08i)T + (-18.8 + 2.20i)T^{2}
23 1+(0.303−0.785i)T+(−17.0−15.4i)T2 1 + (0.303 - 0.785i)T + (-17.0 - 15.4i)T^{2}
29 1+(6.13+4.38i)T+(9.38+27.4i)T2 1 + (6.13 + 4.38i)T + (9.38 + 27.4i)T^{2}
31 1+(6.53−7.49i)T+(−4.19−30.7i)T2 1 + (6.53 - 7.49i)T + (-4.19 - 30.7i)T^{2}
37 1+(−1.20−1.61i)T+(−10.6+35.4i)T2 1 + (-1.20 - 1.61i)T + (-10.6 + 35.4i)T^{2}
41 1+(−0.459+1.78i)T+(−35.8−19.8i)T2 1 + (-0.459 + 1.78i)T + (-35.8 - 19.8i)T^{2}
43 1+(−7.48−6.79i)T+(4.16+42.7i)T2 1 + (-7.48 - 6.79i)T + (4.16 + 42.7i)T^{2}
47 1+(−7.84−8.98i)T+(−6.36+46.5i)T2 1 + (-7.84 - 8.98i)T + (-6.36 + 46.5i)T^{2}
53 1+(−1.65+9.36i)T+(−49.8−18.1i)T2 1 + (-1.65 + 9.36i)T + (-49.8 - 18.1i)T^{2}
59 1+(−2.24+0.916i)T+(42.1−41.3i)T2 1 + (-2.24 + 0.916i)T + (42.1 - 41.3i)T^{2}
61 1+(12.8−7.73i)T+(28.4−53.9i)T2 1 + (12.8 - 7.73i)T + (28.4 - 53.9i)T^{2}
67 1+(4.14−2.96i)T+(21.6−63.3i)T2 1 + (4.14 - 2.96i)T + (21.6 - 63.3i)T^{2}
71 1+(−1.59−5.32i)T+(−59.3+39.0i)T2 1 + (-1.59 - 5.32i)T + (-59.3 + 39.0i)T^{2}
73 1+(−8.65+2.05i)T+(65.2−32.7i)T2 1 + (-8.65 + 2.05i)T + (65.2 - 32.7i)T^{2}
79 1+(−3.32+3.25i)T+(1.53−78.9i)T2 1 + (-3.32 + 3.25i)T + (1.53 - 78.9i)T^{2}
83 1+(2.33+9.08i)T+(−72.6+40.0i)T2 1 + (2.33 + 9.08i)T + (-72.6 + 40.0i)T^{2}
89 1+(−4.90+16.3i)T+(−74.3−48.9i)T2 1 + (-4.90 + 16.3i)T + (-74.3 - 48.9i)T^{2}
97 1+(9.63+0.374i)T+(96.7+7.51i)T2 1 + (9.63 + 0.374i)T + (96.7 + 7.51i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.56861560719803219490764562702, −9.536369841762595585086687511073, −8.879203722049512055722872049729, −7.51145493544754792606562922828, −6.41098670946179051403423577097, −5.88826706228066277372959826544, −5.23262074667289811195285505174, −4.17475104465701385220616251883, −2.96651396556955424227582158800, −1.82177277366703898019244369253, 1.93850500054828259775257784000, 2.54034867558937658111935805561, 3.93481047098660012093734474465, 4.99457764878144650965771630219, 5.61320356720926235889269350324, 6.42278651331975177589951303537, 7.37781466217035072600313520636, 9.025842852322310264889837333049, 9.474681546580762267896399423773, 10.69159912717008772031569053577

Graph of the ZZ-function along the critical line